How to understand the evolution of cooperation remains a scientific challenge. Individual strategy update rule plays an important role in the evolution of cooperation in a population. Previous works mainly assume that individuals adopt one single update rule during the evolutionary process. Indeed, individuals may adopt a mixed update rule influenced by different preferences such as payoff-driven and conformity-driven factors. It is still unclear how such mixed update rules influence the evolutionary dynamics of cooperation from a theoretical analysis perspective. In this work, in combination with the pairwise comparison rule and the conformity rule, we consider a mixed updating procedure into the evolutionary prisoner’s dilemma game. We assume that individuals adopt the conformity rule for strategy updating with a certain probability in a structured population. By means of the pair approximation and mean-field approaches, we obtain the dynamical equations for the fraction of cooperators in the population. We prove that under weak selection, there exists one unique interior equilibrium point, which is stable, in the system. Accordingly, cooperators can survive with defectors under the mixed update rule in the structured population. In addition, we find that the stationary fraction of cooperators increases as the conformity strength increases, but is independent of the benefit parameter. Furthermore, we perform numerical calculations and computer simulations to confirm our theoretical predictions.

1.
W. D.
Hamilton
,
Am. Nat.
97
,
354
356
(
1963
).
4.
M.
Perc
,
J. J.
Jordan
,
D. G.
Rand
et al.,
Phys. Rep.
687
,
1
51
(
2017
).
5.
A. S.
Griffin
,
S. A.
West
, and
A.
Buckling
,
Nature
430
,
1024
1027
(
2004
).
6.
J.
Hofbauer
and
K.
Sigmund
,
Evolutionary Games and Population Dynamics
(
Cambridge University Press
,
1998
).
7.
K.
Jensen
,
J.
Call
, and
M.
Tomasello
,
Science
318
,
107
109
(
2007
).
8.
A.
Rapoport
,
Prisoner’s Dilemma
(
Springer
,
1989
).
9.
M. A.
Nowak
and
K.
Sigmund
,
Nature
364
,
56
58
(
1993
).
10.
P.
Erdős
and
A.
Rényi
,
Publ. Math. Debrecen
6
,
290
297
(
1959
).
11.
M.
Perc
,
J.
Gómez-Gardenes
,
A.
Szolnoki
et al.,
J. R. Soc. Interface
10
,
20120997
(
2013
).
12.
B.
Allen
,
G.
Lippner
,
Y. T.
Chen
et al.,
Nature
544
,
227
230
(
2017
).
13.
M. A.
Nowak
and
R. M.
May
,
Nature
359
,
826
829
(
1992
).
14.
F.
Fu
,
L.
Wang
,
M. A.
Nowak
et al.,
Phys. Rev. E
79
,
046707
(
2009
).
15.
G.
Szabó
and
G.
Fáth
,
Phys. Rep.
446
,
97
216
(
2007
).
16.
Z.
Rong
,
Z.-X.
Wu
,
X.
Li
et al.,
Chaos
29
,
103103
(
2019
).
17.
H.
Ohtsuki
,
C.
Hauert
,
E.
Lieberman
et al.,
Nature
441
,
502
505
(
2006
).
18.
H.
Ohtsuki
and
M. A.
Nowak
,
J. Theor. Biol.
243
,
86
97
(
2006
).
19.
A.
Szolnoki
and
M.
Perc
,
J. R. Soc. Interface
12
,
20141299
(
2015
).
20.
G.
Szabó
and
C.
Tőke
,
Phys. Rev. E
58
,
69
(
1998
).
21.
F.
Fu
,
D. I.
Rosenbloom
,
L.
Wang
et al.,
Proc. R. Soc. B
278
,
42
49
(
2011
).
22.
A.
Traulsen
,
J. M.
Pacheco
, and
M. A.
Nowak
,
J. Theor. Biol.
246
,
522
529
(
2007
).
23.
G.
Abramson
and
M.
Kuperman
,
Phys. Rev. E
63
,
030901
(
2001
).
24.
G.
Szabó
,
A.
Szolnoki
,
M.
Varga
et al.,
Phys. Rev. E
82
,
026110
(
2010
).
25.
K.
Shigaki
,
J.
Tanimoto
,
Z.
Wang
et al.,
Phys. Rev. E
86
,
031141
(
2012
).
26.
J.
Pena
,
E.
Pestelacci
,
M.
Tomassini
et al.,
IEEE Cong. Evol. Comp.
1–5
,
506
(
2009
).
27.
J.
Pena
,
H.
Volken
,
E.
Pestelacci
et al.,
Phys. Rev. E
80
,
016110
(
2009
).
28.
A.
Szolnoki
and
M.
Perc
,
Sci. Rep.
6
,
23633
(
2016
).
29.
Z.
Niu
,
J.
Xu
,
D.
Dai
et al.,
Chaos Soliton. Fract.
112
,
92
96
(
2018
).
30.
X.
Liu
,
C.
Huang
,
Q.
Dai
et al.,
Europhys. Lett.
128
,
18001
(
2019
).
31.
J.
Zhou
,
C.
Huang
, and
Q.
Dai
,
Europhys. Lett.
123
,
30004
(
2018
).
32.
Z.
Yang
,
Z.
Li
, and
L.
Wang
,
Appl. Math. Comput.
379
,
125251
(
2020
).
33.
H.
Chen
and
E. H.
Yong
,
Phys. Rev. E
104
,
014301
(
2021
).
34.
L.
Cremene
and
M.
Cremene
,
Chaos Soliton. Fract.
144
,
110710
(
2021
).
35.
T.
An
,
J.
Wang
,
B.
Zhou
et al.,
Front. Phys.
10
,
972457
(
2022
).
36.
M. A.
Habib
,
M.
Tanaka
, and
J.
Tanimoto
,
Chaos Soliton. Fract.
138
,
109997
(
2020
).
37.
F.
Shu
,
Y.
Liu
,
X.
Liu
et al.,
Appl. Math. Comput.
346
,
480
490
(
2019
).
38.
J.
Lin
,
C.
Huang
,
Q.
Dai
et al.,
Chaos Soliton. Fract.
140
,
110146
(
2020
).
39.
A.
Szolnoki
and
X.
Chen
,
New J. Phys.
20
,
093008
(
2018
).
40.
L.
Zhang
,
C.
Huang
,
H.
Li
et al.,
Physica A
561
,
125260
(
2021
).
41.
J. A. R.
Marshall
,
J. Theor. Biol.
260
,
386
391
(
2009
).
42.
C.
Hilbe
,
M. A.
Nowak
, and
K.
Sigmund
,
Proc. Natl. Acad. Sci.
110
,
6913
6918
(
2013
).
43.
M. A.
Nowak
,
A.
Sasaki
,
C.
Taylor
, and
D.
Fudenberg
,
Nature
428
,
646
650
(
2004
).
44.
H. A.
Gutowitz
,
J. D.
Victor
, and
B. W.
Knight
,
Physica D
28
,
18
48
(
1987
).
45.
G.
Szabó
,
A.
Szolnoki
, and
L.
Bodócs
,
Phys. Rev. A
44
,
6375
6378
(
1991
).
46.
G.
Szabó
,
J.
Vukov
, and
A.
Szolnoki
,
Phys. Rev. E
72
,
047107
(
2005
).
47.
J.
Vukov
,
G.
Szabó
, and
A.
Szolnoki
,
Phys. Rev. E
73
,
067103
(
2006
).
48.
A.
Li
,
M.
Broom
,
J.
Du
et al.,
Phys. Rev. E
93
,
022407
(
2016
).
49.
S.
Wang
,
X.
Chen
,
Z.
Xiao
et al.,
Appl. Math. Comput.
431
,
127308
(
2022
).
50.
S.
Wang
,
X.
Chen
,
Z.
Xiao
et al.,
J. R. Soc. Interface
20
,
20220653
(
2023
).
51.
Z.
Sun
,
X.
Chen
, and
A.
Szolnoki
,
IEEE Trans. Netw. Sci. Eng.
10
,
3975
3988
(
2023
).
52.
M. A.
Amaral
and
M. A.
Javarone
,
Phys. Rev. E
97
,
042305
(
2018
).
53.
H. K.
Khalil
,
Nonlinear Systems
(
Prentice-Hall
,
Englewood Cliffs, NJ
,
1996
).
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